theory Concl_Pres_Clarification
imports Main
begin
  text ‹Clarification and clarsimp that preserve the structure of 
    the subgoal's conclusion, i.e., neither solve it, nor swap it 
    with premises, as, eg, @{thm [source] notE} does.
    ›

  ML ‹
    local 
      open Classical

      fun is_cp_brl (is_elim,thm) = let
        val prems = Thm.prems_of thm
        val nprems = length prems
        val concl = Thm.concl_of thm
      in
        (if is_elim then nprems=2 else nprems=1) andalso let
          val lprem_concl = hd (rev prems)
            |> Logic.strip_assums_concl
        in
          concl aconv lprem_concl
        end
      end

      val not_elim = @{thm notE}
      val hyp_subst_tacs = [Hypsubst.hyp_subst_tac]

      fun eq_contr_tac ctxt i = ematch_tac ctxt [not_elim] i THEN eq_assume_tac i;
      fun eq_assume_contr_tac ctxt = eq_assume_tac ORELSE' eq_contr_tac ctxt;

      fun cp_bimatch_from_nets_tac ctxt =
        biresolution_from_nets_tac ctxt (order_list o filter (is_cp_brl o snd)) true;


    in
      fun cp_clarify_step_tac ctxt =
        let val {safep_netpair, ...} = (rep_cs o claset_of) ctxt in
          appSWrappers ctxt
           (FIRST'
             [eq_assume_contr_tac ctxt,
              FIRST' (map (fn tac => tac ctxt) hyp_subst_tacs),
              cp_bimatch_from_nets_tac ctxt safep_netpair
              ])
        end;
      
        fun cp_clarify_tac ctxt = SELECT_GOAL (REPEAT_DETERM (cp_clarify_step_tac ctxt 1));

        fun cp_clarsimp_tac ctxt =
          Simplifier.safe_asm_full_simp_tac ctxt THEN_ALL_NEW
          cp_clarify_tac (addSss ctxt);


    end


  ›

  method_setup cp_clarify = ‹
    (Classical.cla_method' (CHANGED_PROP oo cp_clarify_tac))
    ›

  method_setup cp_clarsimp = ‹let
    fun clasimp_method' tac =
      Method.sections clasimp_modifiers >> K (SIMPLE_METHOD' o tac);
  in
    clasimp_method' (CHANGED_PROP oo cp_clarsimp_tac)
  end›



end

theory Named_Theorems_Rev 
imports Main
keywords "named_theorems_rev" :: thy_decl
begin

ML ‹

signature NAMED_THEOREMS_REV =
sig
  val member: Proof.context -> string -> thm -> bool
  val get: Proof.context -> string -> thm list
  val add_thm: string -> thm -> Context.generic -> Context.generic
  val del_thm: string -> thm -> Context.generic -> Context.generic
  val add: string -> attribute
  val del: string -> attribute
  val check: Proof.context -> string * Position.T -> string
  val declare: binding -> string -> local_theory -> string * local_theory
end;

structure Named_Theorems_Rev: NAMED_THEOREMS_REV =
struct

(* context data *)

structure Data = Generic_Data
(
  type T = thm Item_Net.T Symtab.table;
  val empty: T = Symtab.empty;
  val extend = I;
  val merge : T * T -> T = Symtab.join (K Item_Net.merge);
);

fun new_entry name =
  Data.map (fn data =>
    if Symtab.defined data name
    then error ("Duplicate declaration of named theorems: " ^ quote name)
    else Symtab.update (name, Thm.full_rules) data);

fun undeclared name = "Undeclared named theorems " ^ quote name;

fun the_entry context name =
  (case Symtab.lookup (Data.get context) name of
    NONE => error (undeclared name)
  | SOME entry => entry);

fun map_entry name f context =
  (the_entry context name; Data.map (Symtab.map_entry name f) context);


(* maintain content *)

fun member ctxt = Item_Net.member o the_entry (Context.Proof ctxt);

fun content context = Item_Net.content o the_entry context;
val get = content o Context.Proof;

fun add_thm name = map_entry name o Item_Net.update;
fun del_thm name = map_entry name o Item_Net.remove;

val add = Thm.declaration_attribute o add_thm;
val del = Thm.declaration_attribute o del_thm;


(* check *)

fun check ctxt (xname, pos) =
  let
    val context = Context.Proof ctxt;
    val fact_ref = Facts.Named ((xname, Position.none), NONE);
    fun err () = error (undeclared xname ^ Position.here pos);
  in
    (case try (Proof_Context.get_fact_generic context) fact_ref of
      SOME (SOME name, _) => if can (the_entry context) name then name else err ()
    | _ => err ())
  end;


(* declaration *)

fun declare binding descr lthy =
  let
    val name = Local_Theory.full_name lthy binding;
    val description =
      "declaration of " ^ (if descr = "" then Binding.name_of binding ^ " rules" else descr);
    val lthy' = lthy
      |> Local_Theory.background_theory (Context.theory_map (new_entry name))
      |> Local_Theory.map_contexts (K (Context.proof_map (new_entry name)))
      |> Local_Theory.add_thms_dynamic (binding, fn context => content context name)
      |> Attrib.local_setup binding (Attrib.add_del (add name) (del name)) description
  in (name, lthy') end;

val _ =
  Outer_Syntax.local_theory @{command_keyword named_theorems_rev}
    "declare named collection of theorems"
    (Parse.and_list1 (Parse.binding -- Scan.optional Parse.text "") >>
      fold (fn (b, descr) => snd o declare b descr));


(* ML antiquotation *)

val _ = Theory.setup
  (ML_Antiquotation.inline @{binding named_theorems_rev}
    (Args.context -- Scan.lift Args.name_position >>
      (fn (ctxt, name) => ML_Syntax.print_string (check ctxt name))));

end;
›

end

theory Pf_Add
imports Automatic_Refinement.Misc "HOL-Library.Monad_Syntax"
begin

lemma fun_ordI:
  assumes "⋀x. ord (f x) (g x)"
  shows "fun_ord ord f g"
  using assms unfolding fun_ord_def by auto

lemma fun_ordD:
  assumes "fun_ord ord f g"
  shows "ord (f x) (g x)"
  using assms unfolding fun_ord_def by auto

lemma mono_fun_fun_cnv:
  assumes "⋀d. monotone (fun_ord ordA) ordB (λx. F x d)"
  shows "monotone (fun_ord ordA) (fun_ord ordB) F"
  apply rule
  apply (rule fun_ordI)
  using assms
  by (blast dest: monotoneD)

lemma fun_lub_Sup[simp]: "fun_lub Sup = Sup"
  unfolding fun_lub_def[abs_def]
  by (clarsimp intro!: ext; metis image_def)

lemma fun_ord_le[simp]: "fun_ord (≤) = (≤)"
  unfolding fun_ord_def[abs_def]
  by (auto intro!: ext simp: le_fun_def)

end

section ‹Interfacing Partial-Function's Monotonicity Prover›
theory Pf_Mono_Prover
imports Separation_Logic_Imperative_HOL.Sep_Main
begin
  (* TODO: Adjust mono-prover accordingly  *)
  (* Wraps mono-prover of partial-function to erase premises. 
    This is a workaround for mono_tac, which does not accept premises if the case-split rule is applied. *)

ML ‹
  structure Pf_Mono_Prover = struct
    fun mono_tac ctxt = (REPEAT o eresolve_tac ctxt @{thms thin_rl})
      THEN' Partial_Function.mono_tac ctxt
  end
›

method_setup pf_mono = ‹Scan.succeed (fn ctxt => SIMPLE_METHOD' (Pf_Mono_Prover.mono_tac ctxt))› ‹Monotonicity prover of the partial function package›

end

theory PO_Normalizer
imports Automatic_Refinement.Refine_Lib
begin
  ML_file ‹PO_Normalizer.ML›
end

signature PO_NORMALIZER = sig 
  type norm_set = {
    trans_rules : thm list, (* Transitivity rules, of form "R x y ⟹ R y z ⟹ R x z" *)
    cong_rules : thm list, (* Congruence rules, of form: "⟦ R1 a1 b1; ... ⟧ ⟹ R (f a1 ...) (f b1 ...)" *)
    norm_rules : thm list, (* Normalization rules, of form: "R f g" *)
    refl_rules : thm list (* Reflexivity rules, of form: "R x x"*)
  }

  val gen_norm_tac : norm_set -> Proof.context -> tactic'
  val gen_norm_rule : thm list -> norm_set -> Proof.context -> thm -> thm
end

structure PO_Normalizer : PO_NORMALIZER = struct
  type norm_set = {
    trans_rules : thm list, (* Transitivity rules, of form "R x y ⟹ R y z ⟹ R x z" *)
    cong_rules : thm list, (* Congruence rules, of form: "⟦ R1 a1 b1; ... ⟧ ⟹ R (f a1 ...) (f b1 ...)" *)
    norm_rules : thm list, (* Normalization rules, of form: "R f g" *)
    refl_rules : thm list (* Reflexivity rules, of form: "R x x"*)
  }

  val cfg_trace = 
    Attrib.setup_config_bool @{binding "norm_rel_trace"} (K false)

  val cfg_depth_limit = 
    Attrib.setup_config_int @{binding "norm_rel_depth_limit"} (K ~1)


  fun gen_norm_tac {trans_rules, cong_rules, norm_rules, refl_rules} ctxt = let
    val do_trace = Config.get ctxt cfg_trace

    fun trace_tac str _ st = if do_trace then 
      (tracing str; Seq.single st)
    else Seq.single st
    val print_tac = if do_trace then print_tac else (K (K all_tac))

    val depth_limit = Config.get ctxt cfg_depth_limit

    fun norm_tac d ctxt i st = let
      val transr_tac = resolve_tac ctxt trans_rules
      val congr_tac = resolve_tac ctxt cong_rules
      val rewrr_tac = resolve_tac ctxt norm_rules
      val solver_tac = resolve_tac ctxt refl_rules

      val cong_tac = (transr_tac THEN' (
        (congr_tac THEN' trace_tac "cong") THEN_ALL_NEW_FWD norm_tac (d+1) ctxt))
      val rewr_tac = (transr_tac THEN' (SOLVED' rewrr_tac) 
        THEN' trace_tac "rewr" THEN' transr_tac THEN' norm_tac (d+1) ctxt)
      val solve_tac = SOLVED' solver_tac THEN' (K (print_tac ctxt "solved"))
    in 
      if depth_limit>=0 andalso d>depth_limit then
        (K (print_tac ctxt "Norm-Depth limit reached"))
        THEN' solve_tac
      else
        (K (print_tac ctxt ("Normalizing ("^ string_of_int d  ^")"))) THEN'
        (TRY o cong_tac)
        THEN' (TRY o rewr_tac)
        THEN' solve_tac
    end i st
  in norm_tac 1 ctxt end

  fun gen_norm_rule init_thms norm_set ctxt thm = let
    val orig_ctxt = ctxt
    val ((_,[thm]),ctxt) = Variable.import false [thm] ctxt

    fun tac ctxt = 
      eresolve_tac ctxt init_thms
      THEN' gen_norm_tac norm_set ctxt

    val concl = Thm.concl_of thm
    val x = Var (("x",0),@{typ prop})
    val t = @{mk_term "PROP ?concl ⟹ PROP ?x"}

    val thm2 = Goal.prove ctxt [] [] t 
      (fn {context = ctxt, ...} => tac ctxt 1)
    
    val thm = thm RS thm2 
    val [thm] = Variable.export ctxt orig_ctxt [thm]
  in
    thm
  end
  
end

theory Sepref_Misc
imports 
  Refine_Monadic.Refine_Monadic
  PO_Normalizer
  "List-Index.List_Index"
  Separation_Logic_Imperative_HOL.Sep_Main
  Named_Theorems_Rev
  "HOL-Eisbach.Eisbach"
  Separation_Logic_Imperative_HOL.Array_Blit
begin

  hide_const (open) CONSTRAINT

  (* Additions for List_Index *)  
  lemma index_of_last_distinct[simp]: 
    "distinct l ⟹ index l (last l) = length l - 1"  
    apply (cases l rule: rev_cases)
    apply (auto simp: index_append)
    done

  lemma index_eqlen_conv[simp]: "index l x = length l ⟷ x∉set l"
    by (auto simp: index_size_conv)


  subsection ‹Iterated Curry and Uncurry›    


  text ‹Uncurry0›  
  definition "uncurry0 c ≡ λ_::unit. c"
  definition curry0 :: "(unit ⇒ 'a) ⇒ 'a" where "curry0 f = f ()"
  lemma uncurry0_apply[simp]: "uncurry0 c x = c" by (simp add: uncurry0_def)

  lemma curry_uncurry0_id[simp]: "curry0 (uncurry0 f) = f" by (simp add: curry0_def)
  lemma uncurry_curry0_id[simp]: "uncurry0 (curry0 g) = g" by (auto simp: curry0_def)
  lemma param_uncurry0[param]: "(uncurry0,uncurry0) ∈ A → (unit_rel→A)" by auto
    
  text ‹Abbreviations for higher-order uncurries›    
  abbreviation "uncurry2 f ≡ uncurry (uncurry f)"
  abbreviation "curry2 f ≡ curry (curry f)"
  abbreviation "uncurry3 f ≡ uncurry (uncurry2 f)"
  abbreviation "curry3 f ≡ curry (curry2 f)"
  abbreviation "uncurry4 f ≡ uncurry (uncurry3 f)"
  abbreviation "curry4 f ≡ curry (curry3 f)"
  abbreviation "uncurry5 f ≡ uncurry (uncurry4 f)"
  abbreviation "curry5 f ≡ curry (curry4 f)"
  abbreviation "uncurry6 f ≡ uncurry (uncurry5 f)"
  abbreviation "curry6 f ≡ curry (curry5 f)"
  abbreviation "uncurry7 f ≡ uncurry (uncurry6 f)"
  abbreviation "curry7 f ≡ curry (curry6 f)"
  abbreviation "uncurry8 f ≡ uncurry (uncurry7 f)"
  abbreviation "curry8 f ≡ curry (curry7 f)"
  abbreviation "uncurry9 f ≡ uncurry (uncurry8 f)"
  abbreviation "curry9 f ≡ curry (curry8 f)"

    
    
  lemma fold_partial_uncurry: "uncurry (λ(ps, cf). f ps cf) = uncurry2 f" by auto

  lemma curry_shl: 
    "⋀g f. (g ≡ curry f) ≡ (uncurry g ≡ f)"
    "⋀g f. (g ≡ curry0 f) ≡ (uncurry0 g ≡ f)"
    by (atomize (full); auto)+
  
  lemma curry_shr: 
    "⋀f g. (curry f ≡ g) ≡ (f ≡ uncurry g)"
    "⋀f g. (curry0 f ≡ g) ≡ (f ≡ uncurry0 g)"
    by (atomize (full); auto)+
  
  lemmas uncurry_shl = curry_shr[symmetric]  
  lemmas uncurry_shr = curry_shl[symmetric]  
  
end

section ‹Subgoal Structure for Apply Scripts›
theory Structured_Apply
imports Main
keywords 
  "focus" "solved" "applyS" "apply1" "applyF" "applyT" :: prf_script
begin

text ‹This theory provides some variants of the apply command 
  that make the proof structure explicit. See below for examples.

  Compared to the @{command subgoal}-command, these set of commands is more lightweight,
  and fully supports schematic variables.
›

(*
  focus, focus <method text>, applyF <method text>
    Focus on current subgoal, and then (optionally) apply method. applyF m is a synonym for focus m.

  solved
    Assert that subgoal is solved and release focus.

  applyT <method text>
    Apply method to current subgoal only. Same as apply m [].

  applyS <method text>
    Apply method to current subgoal, and assert that subgoal is solved.
    "applyS m" is roughly equal to "focus m solved"

  apply1 <method text>
    Apply method to current subgoal, and assert that there is exactly one resulting subgoal.

*)

ML ‹
signature STRUCTURED_APPLY = sig
  val focus: Proof.state -> Proof.state
  val solved: Proof.state -> Proof.state
  val unfocus: Proof.state -> Proof.state

  val apply1: Method.text_range -> Proof.state -> Proof.state Seq.result Seq.seq
  val applyT: Method.text * Position.range -> Proof.state -> Proof.state Seq.result Seq.seq
  val apply_focus: Method.text_range -> Proof.state -> Proof.state Seq.result Seq.seq
  val apply_solve: Method.text_range -> Proof.state -> Proof.state Seq.result Seq.seq
end

structure Structured_Apply: STRUCTURED_APPLY = struct
  val focus = Proof.refine_primitive (K (Goal.restrict 1 1))
  val unfocus = Proof.refine_primitive (K (Goal.unrestrict 1))
  val solved = Proof.refine_primitive (fn _ => fn thm => let
      val _ = if Thm.nprems_of thm > 0 then error "Subgoal not solved" else ()
    in
      Goal.unrestrict 1 thm
    end
  )

  fun apply_focus m = focus #> Proof.apply m

  fun assert_num_solved d msg m s = let
    val n_subgoals = Proof.raw_goal #> #goal #> Thm.nprems_of
    val n1 = n_subgoals s

    fun do_assert s = if n1 - n_subgoals s <> d then error msg else s
  in
    s 
    |> Proof.apply m
    |> Seq.map_result do_assert
  end

  fun apply_solve m = 
      focus 
    #> assert_num_solved 1 "Subgoal not solved" m
    #> Seq.map_result unfocus

  fun apply1 m = 
      focus 
    #> assert_num_solved 0 "Method must not produce or solve subgoals" m 
    #> Seq.map_result unfocus

  fun applyT (m,pos) = let
    open Method
    val m = Combinator (no_combinator_info, Select_Goals 1, [m])
  in
    Proof.apply (m,pos)
  end  


end

val _ =
  Outer_Syntax.command @{command_keyword solved} "Primitive unfocus after subgoal is solved"
    (Scan.succeed ( Toplevel.proof (Structured_Apply.solved) ));

val _ =
  Outer_Syntax.command @{command_keyword focus} "Primitive focus then optionally apply method"
    (Scan.option Method.parse >> (fn 
        NONE => Toplevel.proof (Structured_Apply.focus)
      | SOME m => (Method.report m; Toplevel.proofs (Structured_Apply.apply_focus m))
    ));

val _ =
  Outer_Syntax.command @{command_keyword applyF} "Primitive focus then apply method"
    (Method.parse >> (fn m => (Method.report m; 
      Toplevel.proofs (Structured_Apply.apply_focus m)
    )));

val _ =
  Outer_Syntax.command @{command_keyword applyS} "Apply method that solves exactly one subgoal"
    (Method.parse >> (fn m => (Method.report m; 
      Toplevel.proofs (Structured_Apply.apply_solve m) 
    )));

val _ =
  Outer_Syntax.command @{command_keyword apply1} "Apply method that does not change number of subgoals"
    (Method.parse >> (fn m => (Method.report m; 
      Toplevel.proofs (Structured_Apply.apply1 m) 
    )));

val _ =
  Outer_Syntax.command @{command_keyword applyT} "Apply method on first subgoal"
    (Method.parse >> (fn m => (Method.report m; 
      Toplevel.proofs (Structured_Apply.applyT m) 
    )));

›


end

section ‹Rule-Based Synthesis of Terms›
theory Term_Synth
imports Sepref_Misc
begin
  definition SYNTH_TERM :: "'a::{} ⇒ 'b::{} ⇒ bool"
    ― ‹Indicate synthesis of @{term y} from @{term x}.›
    where [simp]: "SYNTH_TERM x y ≡ True"
  consts SDUMMY :: "'a :: {}"
    ― ‹After synthesis has been completed, these are replaced by fresh schematic variable›

  named_theorems_rev synth_rules ‹Term synthesis rules›

  text ‹Term synthesis works by proving @{term "SYNTH_TERM t v"}, by repeatedly applying the 
    first matching intro-rule from ‹synth_rules›.  ›


ML ‹
  signature TERM_SYNTH = sig
    (* Synthesize something from term t. The initial list of theorems is
      added to beginning of synth_rules, and can be used to install intro-rules
      for SYNTH_TERM.*)
    val synth_term: thm list -> Proof.context -> term -> term
  end


  structure Term_Synth : TERM_SYNTH = struct

    (* Assumption: Term does not contain dummy variables *)
    fun replace_sdummies t = let
      fun r (t1$t2) n = let
              val (t1,n) = r t1 n
              val (t2,n) = r t2 n
            in (t1$t2,n) end
        | r (Abs (x,T,t)) n = let
              val (t,n) = r t n
            in (Abs (x,T,t),n) end
        | r @{mpat (typs) "SDUMMY::?'v_T"} n = (Var (("_dummy",n),T),n+1)
        | r (t' as (Var ((name,_),_))) n = if String.isPrefix "_" name then raise TERM ("replace_sdummies: Term already contains dummy patterns",[t',t]) else (t',n)
        | r t n = (t,n)
    in
      fst (r t 0)
    end    

    (* Use synthesis rules to transform the given term *)
    fun synth_term thms ctxt t = let
      val orig_ctxt = ctxt
      val (t,ctxt) = yield_singleton (Variable.import_terms true) t ctxt
      val v = Var (("result",0),TVar (("T",0),[]))
      val goal = @{mk_term "Trueprop (SYNTH_TERM ?t ?v)"} |> Thm.cterm_of ctxt
  
      val rules = thms @ Named_Theorems_Rev.get ctxt @{named_theorems_rev synth_rules}
        |> Tactic.build_net
      fun tac ctxt = ALLGOALS (TRY_SOLVED' (
        REPEAT_DETERM' (CHANGED o resolve_from_net_tac ctxt rules)))
      
      val thm = Goal.prove_internal ctxt [] goal (fn _ => tac ctxt)

      val res = case Thm.concl_of thm of
          @{mpat "Trueprop (SYNTH_TERM _ ?res)"} => res 
        | _ => raise THM("Synth_Term: Proved a different theorem?",~1,[thm])

      val res = singleton (Variable.export_terms ctxt orig_ctxt) res
        |> replace_sdummies
  
    in
      res
    end
  end  
›



end

theory User_Smashing
  imports Pure
begin
(* Alternative flex-flex smasher by Simon Wimmer *)
ML ‹
  fun enumerate xs = fold (fn x => fn (i, xs) => (i +1, (x, i) :: xs)) xs (0, []) |> snd
›

ML ‹
  fun dummy_abs _ [] t = t
    | dummy_abs n (T :: Ts) t = Abs ("x" ^ Int.toString n, T, dummy_abs (n + 1) Ts t)
›

ML ‹
  fun common_prefix Ts (t1 as Abs (_, T, t)) (u1 as Abs (_, U, u)) =
    if U = T then common_prefix (T :: Ts) t u else ([], t1, u1)
    | common_prefix Ts t u = (Ts, t, u);

  fun dest_app acc (t $ u) = dest_app (u :: acc) t
    | dest_app acc t = (t, acc);

  fun add_bound (Bound i, n) bs = (i, n) :: bs
    | add_bound _ bs = bs;
›

ML ‹
  fun smash_pair ctxt thm (t, u) =
    let
      val idx = Thm.maxidx_of thm + 1;
      val ctxt' = ctxt;
      val (Ts, t1, _) = common_prefix [] t u;
      val (tas, t2) = Term.strip_abs t;
      val (uas, u2) = Term.strip_abs u;
      val (tx as Var (_, T1), ts) = Term.strip_comb t2;
      val (ux as Var (_, U1), us) = Term.strip_comb u2;
      val Ts1 = Term.binder_types T1;
      val Us1 = Term.binder_types U1;
      val T = Term.fastype_of1 (Ts, t1);
      val tshift = length tas - length Ts;
      val ushift = length uas - length Ts;
      val tbs = fold add_bound (enumerate (rev ts)) [] |> map (apfst (fn i => i - tshift));
      val ubs = fold add_bound (enumerate (rev us)) [] |> map (apfst (fn i => i - ushift));
      val bounds = inter (op =) (map fst tbs) (map fst ubs) |> distinct (=);
      val T' = map (nth Ts) bounds ---> T;
      val v = Var (("simon", idx), T');
      val tbs' = map (fn i => find_first (fn (j, _) => i = j) tbs |> the |> snd |> Bound) bounds;
      val t' = list_comb (v, tbs') |> dummy_abs 0 Ts1;
      (* Need to add bounds for superfluous abstractions here *)
      val ubs' = map (fn i => find_first (fn (j, _) => i = j) ubs |> the |> snd |> Bound) bounds;
      val u' = list_comb (v, ubs') |> dummy_abs 0 Us1;
      val subst = [(Term.dest_Var tx, Thm.cterm_of ctxt' t'), (Term.dest_Var ux, Thm.cterm_of ctxt' u')];
    in
      instantiate_normalize ([], subst) thm
    end;
    fun smash ctxt thm =
      case (Thm.tpairs_of thm) of
        [] => thm
      | (p :: _) => smash_pair ctxt thm p;
    fun smashed_attrib ctxt thm =
      (NONE, SOME (smash ctxt thm));
›

ML ‹
  val smash_new_rule = Seq.single oo smash;
›

end

chapter ‹The Sepref Tool›
text ‹This chapter contains the Sepref tool and related tools.›
(*<*)
theory Sepref_Chapter_Tool
imports Main
begin
end
(*>*)

section ‹Operation Identification Phase›
theory Sepref_Id_Op
imports 
  Main 
  Automatic_Refinement.Refine_Lib
  Automatic_Refinement.Autoref_Tagging
  "Lib/Named_Theorems_Rev"
begin

text ‹
  The operation identification phase is adapted from the Autoref tool.
  The basic idea is to have a type system, which works on so called 
  interface types (also called conceptual types). Each conceptual type
  denotes an abstract data type, e.g., set, map, priority queue.
  
  Each abstract operation, which must be a constant applied to its arguments,
  is assigned a conceptual type. Additionally, there is a set of 
  {\emph pattern rewrite rules},
  which are applied to subterms before type inference takes place, and 
  which may be backtracked over. 
  This way, encodings of abstract operations in Isabelle/HOL, like 
  @{term [source] "λ_. None"} for the empty map, 
  or @{term [source] "fun_upd m k (Some v)"} for map update, can be rewritten
  to abstract operations, and get properly typed.
›

subsection "Proper Protection of Term"
text ‹ The following constants are meant to encode abstraction and 
  application as proper HOL-constants, and thus avoid strange effects with
  HOL's higher-order unification heuristics and automatic 
  beta and eta-contraction.

  The first step of operation identification is to protect the term
  by replacing all function applications and abstractions be 
  the constants defined below.
›

definition [simp]: "PROTECT2 x (y::prop) ≡ x"
consts DUMMY :: "prop"

abbreviation PROTECT2_syn ("'(#_#')") where "PROTECT2_syn t ≡ PROTECT2 t DUMMY"

abbreviation (input)ABS2 :: "('a⇒'b)⇒'a⇒'b" (binder "λ⇩2" 10)
  where "ABS2 f ≡ (λx. PROTECT2 (f x) DUMMY)"

lemma beta: "(λ⇩2x. f x)$x ≡ f x" by simp

text ‹
  Another version of @{const "APP"}. Treated like @{const APP} by our tool.
  Required to avoid infinite pattern rewriting in some cases, e.g., map-lookup.
›

definition APP' (infixl "$''" 900) where [simp, autoref_tag_defs]: "f$'a ≡ f a"

text ‹
  Sometimes, whole terms should be protected from being processed by our tool.
  For example, our tool should not look into numerals. For this reason,
  the ‹PR_CONST› tag indicates terms that our tool shall handle as
  atomic constants, an never look into them.

  The special form ‹UNPROTECT› can be used inside pattern rewrite rules.
  It has the effect to revert the protection from its argument, and then wrap
  it into a ‹PR_CONST›.
›
definition [simp, autoref_tag_defs]: "PR_CONST x ≡ x" ― ‹Tag to protect constant›
definition [simp, autoref_tag_defs]: "UNPROTECT x ≡ x" ― ‹Gets 
  converted to @{term PR_CONST}, after unprotecting its content›


subsection ‹Operation Identification›

text ‹ Indicator predicate for conceptual typing of a constant ›
definition intf_type :: "'a ⇒ 'b itself ⇒ bool" (infix "::⇩i" 10) where
  [simp]: "c::⇩iI ≡ True"

lemma itypeI: "c::⇩iI" by simp
lemma itypeI': "intf_type c TYPE('T)" by (rule itypeI)

lemma itype_self: "(c::'a) ::⇩i TYPE('a)" by simp

definition CTYPE_ANNOT :: "'b ⇒ 'a itself ⇒ 'b" (infix ":::⇩i" 10) where
  [simp]: "c:::⇩iI ≡ c"

text ‹ Wrapper predicate for an conceptual type inference ›
definition ID :: "'a ⇒ 'a ⇒ 'c itself ⇒ bool" 
  where [simp]: "ID t t' T ≡ t=t'"

subsubsection ‹Conceptual Typing Rules›

lemma ID_unfold_vars: "ID x y T ⟹ x≡y" by simp
lemma ID_PR_CONST_trigger: "ID (PR_CONST x) y T ⟹ ID (PR_CONST x) y T" .

lemma pat_rule:
  "⟦ p≡p'; ID p' t' T ⟧ ⟹ ID p t' T" by simp

lemma app_rule:
  "⟦ ID f f' TYPE('a⇒'b); ID x x' TYPE('a)⟧ ⟹ ID (f$x) (f'$x') TYPE('b)"
  by simp

lemma app'_rule:
  "⟦ ID f f' TYPE('a⇒'b); ID x x' TYPE('a)⟧ ⟹ ID (f$'x) (f'$x') TYPE('b)"
  by simp

lemma abs_rule:
  "⟦ ⋀x x'. ID x x' TYPE('a) ⟹ ID (t x) (t' x x') TYPE('b) ⟧ ⟹
    ID (λ⇩2x. t x) (λ⇩2x'. t' x' x') TYPE('a⇒'b)"
  by simp

lemma id_rule: "c::⇩iI ⟹ ID c c I" by simp

lemma annot_rule: "ID t t' I ⟹ ID (t:::⇩iI) t' I"
  by simp

lemma fallback_rule:
  "ID (c::'a) c TYPE('c)"
  by simp

lemma unprotect_rl1: "ID (PR_CONST x) t T ⟹ ID (UNPROTECT x) t T"
  by simp

subsection ‹ ML-Level code ›
ML ‹
infix 0 THEN_ELSE_COMB'

signature ID_OP_TACTICAL = sig
  val SOLVE_FWD: tactic' -> tactic'
  val DF_SOLVE_FWD: bool -> tactic' -> tactic'
end

structure Id_Op_Tactical :ID_OP_TACTICAL = struct

  fun SOLVE_FWD tac i st = SOLVED' (
    tac 
    THEN_ALL_NEW_FWD (SOLVE_FWD tac)) i st


  (* Search for solution with DFS-strategy. If dbg-flag is given,
    return sequence of stuck states if no solution is found.
  *)
  fun DF_SOLVE_FWD dbg tac = let
    val stuck_list_ref = Unsynchronized.ref []

    fun stuck_tac _ st = if dbg then (
      stuck_list_ref := st :: !stuck_list_ref;
      Seq.empty
    ) else Seq.empty

    fun rec_tac i st = (
        (tac THEN_ALL_NEW_FWD (SOLVED' rec_tac))
        ORELSE' stuck_tac
      ) i st

    fun fail_tac _ _ = if dbg then
      Seq.of_list (rev (!stuck_list_ref))
    else Seq.empty
  in
    rec_tac ORELSE' fail_tac    
  end

end
›


named_theorems_rev id_rules "Operation identification rules"
named_theorems_rev pat_rules "Operation pattern rules"
named_theorems_rev def_pat_rules "Definite operation pattern rules (not backtracked over)"



ML ‹

  structure Id_Op = struct

    fun id_a_conv cnv ct = case Thm.term_of ct of
      @{mpat "ID _ _ _"} => Conv.fun_conv (Conv.fun_conv (Conv.arg_conv cnv)) ct
    | _ => raise CTERM("id_a_conv",[ct])

    fun 
      protect env (@{mpat "?t:::⇩i?I"}) = let
        val t = protect env t
      in 
        @{mk_term env: "?t:::⇩i?I"}
      end
    | protect _ (t as @{mpat "PR_CONST _"}) = t
    | protect env (t1$t2) = let
        val t1 = protect env t1
        val t2 = protect env t2
      in
        @{mk_term env: "?t1.0 $ ?t2.0"}
      end
    | protect env (Abs (x,T,t)) = let
        val t = protect (T::env) t
      in
        @{mk_term env: "λv_x::?'v_T. PROTECT2 ?t DUMMY"}
      end
    | protect _ t = t

    fun protect_conv ctxt = Refine_Util.f_tac_conv ctxt
      (protect []) 
      (simp_tac 
        (put_simpset HOL_basic_ss ctxt addsimps @{thms PROTECT2_def APP_def}) 1)

    fun unprotect_conv ctxt
      = Simplifier.rewrite (put_simpset HOL_basic_ss ctxt 
        addsimps @{thms PROTECT2_def APP_def})

    fun do_unprotect_tac ctxt =
      resolve_tac ctxt @{thms unprotect_rl1} THEN'
      CONVERSION (Refine_Util.HOL_concl_conv (fn ctxt => id_a_conv (unprotect_conv ctxt)) ctxt)

    val cfg_id_debug = 
      Attrib.setup_config_bool @{binding id_debug} (K false)

    val cfg_id_trace_fallback = 
      Attrib.setup_config_bool @{binding id_trace_fallback} (K false)

    fun dest_id_rl thm = case Thm.concl_of thm of
      @{mpat (typs) "Trueprop (?c::⇩iTYPE(?'v_T))"} => (c,T)
    | _ => raise THM("dest_id_rl",~1,[thm])

    
    val add_id_rule = snd oo Thm.proof_attributes [Named_Theorems_Rev.add @{named_theorems_rev id_rules}]

    datatype id_tac_mode = Init | Step | Normal | Solve

    fun id_tac ss ctxt = let
      open Id_Op_Tactical
      val certT = Thm.ctyp_of ctxt
      val cert = Thm.cterm_of ctxt

      val thy = Proof_Context.theory_of ctxt

      val id_rules = Named_Theorems_Rev.get ctxt @{named_theorems_rev id_rules}
      val pat_rules = Named_Theorems_Rev.get ctxt @{named_theorems_rev pat_rules}
      val def_pat_rules = Named_Theorems_Rev.get ctxt @{named_theorems_rev def_pat_rules}

      val rl_net = Tactic.build_net (
        (pat_rules |> map (fn thm => thm RS @{thm pat_rule})) 
        @ @{thms annot_rule app_rule app'_rule abs_rule} 
        @ (id_rules |> map (fn thm => thm RS @{thm id_rule}))
      )

      val def_rl_net = Tactic.build_net (
        (def_pat_rules |> map (fn thm => thm RS @{thm pat_rule}))
      )  

      val id_pr_const_rename_tac = 
          resolve_tac ctxt @{thms ID_PR_CONST_trigger} THEN'
          Subgoal.FOCUS (fn { context=ctxt, prems, ... } => 
            let
              fun is_ID @{mpat "Trueprop (ID _ _ _)"} = true | is_ID _ = false
              val prems = filter (Thm.prop_of #> is_ID) prems
              val eqs = map (fn thm => thm RS @{thm ID_unfold_vars}) prems
              val conv = Conv.rewrs_conv eqs
              val conv = fn ctxt => (Conv.top_sweep_conv (K conv) ctxt)
              val conv = fn ctxt => Conv.fun2_conv (Conv.arg_conv (conv ctxt))
              val conv = Refine_Util.HOL_concl_conv conv ctxt
            in CONVERSION conv 1 end 
          ) ctxt THEN'
          resolve_tac ctxt @{thms id_rule} THEN'
          resolve_tac ctxt id_rules 

      val ityping = id_rules 
        |> map dest_id_rl
        |> filter (is_Const o #1)
        |> map (apfst (#1 o dest_Const))
        |> Symtab.make_list

      val has_type = Symtab.defined ityping

      fun mk_fallback name cT =
        case try (Sign.the_const_constraint thy) name of
          SOME T => try (Thm.instantiate' 
                          [SOME (certT cT), SOME (certT T)] [SOME (cert (Const (name,cT)))])
                        @{thm fallback_rule} 
        | NONE => NONE

      fun trace_fallback thm = 
        Config.get ctxt cfg_id_trace_fallback       
        andalso let 
          open Pretty
          val p = block [str "ID_OP: Applying fallback rule: ", Thm.pretty_thm ctxt thm]
        in 
          string_of p |> tracing; 
          false
        end  

      val fallback_tac = CONVERSION Thm.eta_conversion THEN' IF_EXGOAL (fn i => fn st =>
        case Logic.concl_of_goal (Thm.prop_of st) i of
          @{mpat "Trueprop (ID (mpaq_STRUCT (mpaq_Const ?name ?cT)) _ _)"} => (
            if not (has_type name) then 
              case mk_fallback name cT of
                SOME thm => (trace_fallback thm; resolve_tac ctxt [thm] i st)
              | NONE => Seq.empty  
            else Seq.empty
          )
        | _ => Seq.empty)

      val init_tac = CONVERSION (
        Refine_Util.HOL_concl_conv (fn ctxt => (id_a_conv (protect_conv ctxt))) 
          ctxt
      )

      val step_tac = (FIRST' [
        assume_tac ctxt, 
        eresolve_tac ctxt @{thms id_rule},
        resolve_from_net_tac ctxt def_rl_net, 
        resolve_from_net_tac ctxt rl_net, 
        id_pr_const_rename_tac,
        do_unprotect_tac ctxt, 
        fallback_tac])

      val solve_tac = DF_SOLVE_FWD (Config.get ctxt cfg_id_debug) step_tac  

    in
      case ss of
        Init => init_tac 
      | Step => step_tac 
      | Normal => init_tac THEN' solve_tac
      | Solve => solve_tac

    end

  end

›

subsection ‹Default Setup›

subsubsection ‹Numerals› 
lemma pat_numeral[def_pat_rules]: "numeral$x ≡ UNPROTECT (numeral$x)" by simp

lemma id_nat_const[id_rules]: "(PR_CONST (a::nat)) ::⇩i TYPE(nat)" by simp
lemma id_int_const[id_rules]: "(PR_CONST (a::int)) ::⇩i TYPE(int)" by simp

(*subsection ‹Example›
schematic_lemma 
  "ID (λa b. (b(1::int↦2::nat) |`(-{3})) a, Map.empty, λa. case a of None ⇒ Some a | Some _ ⇒ None) (?c) (?T::?'d itself)"
  (*"TERM (?c,?T)"*)
  using [[id_debug]]
  apply (tactic {* Id_Op.id_tac Id_Op.Normal @{context} 1  *})  
  done
*)

end

section ‹Basic Definitions›
theory Sepref_Basic
imports 
  "HOL-Eisbach.Eisbach"
  Separation_Logic_Imperative_HOL.Sep_Main
  Refine_Monadic.Refine_Monadic
  "Lib/Sepref_Misc"
  "Lib/Structured_Apply"
  Sepref_Id_Op
begin
no_notation i_ANNOT (infixr ":::⇩i" 10)
no_notation CONST_INTF (infixr "::⇩i" 10)

text ‹
  In this theory, we define the basic concept of refinement 
  from a nondeterministic program specified in the 
  Isabelle Refinement Framework to an imperative deterministic one 
  specified in Imperative/HOL.
›

subsection ‹Values on Heap›
text ‹We tag every refinement assertion with the tag ‹hn_ctxt›, to
  avoid higher-order unification problems when the refinement assertion 
  is schematic.›
definition hn_ctxt :: "('a⇒'c⇒assn) ⇒ 'a ⇒ 'c ⇒ assn" 
  ― ‹Tag for refinement assertion›
  where
  "hn_ctxt P a c ≡ P a c"

definition pure :: "('b × 'a) set ⇒ 'a ⇒ 'b ⇒ assn"
  ― ‹Pure binding, not involving the heap›
  where "pure R ≡ (λa c. ↑((c,a)∈R))"

lemma pure_app_eq: "pure R a c = ↑((c,a)∈R)" by (auto simp: pure_def)

lemma pure_eq_conv[simp]: "pure R = pure R' ⟷ R=R'"
  unfolding pure_def 
  apply (rule iffI)
  apply safe
  apply (meson pure_assn_eq_conv)
  apply (meson pure_assn_eq_conv)
  done

lemma pure_rel_eq_false_iff: "pure R x y = false ⟷ (y,x)∉R"
  by (auto simp: pure_def)
    
    
definition "is_pure P ≡ ∃P'. ∀x x'. P x x'=↑(P' x x')"
lemma is_pureI[intro?]: 
  assumes "⋀x x'. P x x' = ↑(P' x x')"
  shows "is_pure P"
  using assms unfolding is_pure_def by blast

lemma is_pureE:
  assumes "is_pure P"
  obtains P' where "⋀x x'. P x x' = ↑(P' x x')"
  using assms unfolding is_pure_def by blast

lemma pure_pure[simp]: "is_pure (pure P)"
  unfolding pure_def by rule blast
lemma pure_hn_ctxt[intro!]: "is_pure P ⟹ is_pure (hn_ctxt P)"
  unfolding hn_ctxt_def[abs_def] .


definition "the_pure P ≡ THE P'. ∀x x'. P x x'=↑((x',x)∈P')"

lemma the_pure_pure[simp]: "the_pure (pure R) = R"
  unfolding pure_def the_pure_def
  by (rule theI2[where a=R]) auto

lemma is_pure_alt_def: "is_pure R ⟷ (∃Ri. ∀x y. R x y = ↑((y,x)∈Ri))"
  unfolding is_pure_def
  apply auto
  apply (rename_tac P')
  apply (rule_tac x="{(x,y). P' y x}" in exI)
  apply auto
  done

lemma pure_the_pure[simp]: "is_pure R ⟹ pure (the_pure R) = R"
  unfolding is_pure_alt_def pure_def the_pure_def
  apply (intro ext)
  apply clarsimp
  apply (rename_tac a c Ri)
  apply (rule_tac a=Ri in theI2)
  apply auto
  done
  
lemma is_pure_conv: "is_pure R ⟷ (∃R'. R = pure R')"
  unfolding pure_def is_pure_alt_def by force

lemma is_pure_the_pure_id_eq[simp]: "is_pure R ⟹ the_pure R = Id ⟷ R=pure Id"  
  by (auto simp: is_pure_conv)

lemma is_pure_iff_pure_assn: "is_pure P = (∀x x'. is_pure_assn (P x x'))"
  unfolding is_pure_def is_pure_assn_def by metis



abbreviation "hn_val R ≡ hn_ctxt (pure R)"

lemma hn_val_unfold: "hn_val R a b = ↑((b,a)∈R)"
  by (simp add: hn_ctxt_def pure_def)


definition "invalid_assn R x y ≡ ↑(∃h. h⊨R x y) * true"

abbreviation "hn_invalid R ≡ hn_ctxt (invalid_assn R)"

lemma invalidate_clone: "R x y ⟹⇩A invalid_assn R x y * R x y"
  apply (rule entailsI)
  unfolding invalid_assn_def
  apply (auto simp: models_in_range mod_star_trueI)
  done

lemma invalidate_clone': "R x y ⟹⇩A invalid_assn R x y * R x y * true"
  apply (rule entailsI)
  unfolding invalid_assn_def
  apply (auto simp: models_in_range mod_star_trueI)
  done

lemma invalidate: "R x y ⟹⇩A invalid_assn R x y"
  apply (rule entailsI)
  unfolding invalid_assn_def
  apply (auto simp: models_in_range mod_star_trueI)
  done

lemma invalid_pure_recover: "invalid_assn (pure R) x y = pure R x y * true"
  apply (rule ent_iffI) 
  subgoal
    apply (rule entailsI)
    unfolding invalid_assn_def
    by (auto simp: pure_def)
  subgoal
    unfolding invalid_assn_def
    by (auto simp: pure_def)
  done    

lemma hn_invalidI: "h⊨hn_ctxt P x y ⟹ hn_invalid P x y = true"
  apply (cases h)
  apply (rule ent_iffI)
  apply (auto simp: invalid_assn_def hn_ctxt_def)
  done

lemma invalid_assn_cong[cong]:
  assumes "x≡x'"
  assumes "y≡y'"
  assumes "R x' y' ≡ R' x' y'"
  shows "invalid_assn R x y = invalid_assn R' x' y'"
  using assms unfolding invalid_assn_def
  by simp

subsection ‹Constraints in Refinement Relations›

lemma mod_pure_conv[simp]: "(h,as)⊨pure R a b ⟷ (as={} ∧ (b,a)∈R)"
  by (auto simp: pure_def)

definition rdomp :: "('a ⇒ 'c ⇒ assn) ⇒ 'a ⇒ bool" where
  "rdomp R a ≡ ∃h c. h ⊨ R a c"

abbreviation "rdom R ≡ Collect (rdomp R)"

lemma rdomp_ctxt[simp]: "rdomp (hn_ctxt R) = rdomp R"
  by (simp add: hn_ctxt_def[abs_def])  

lemma rdomp_pure[simp]: "rdomp (pure R) a ⟷ a∈Range R"
  unfolding rdomp_def pure_def by auto

lemma rdom_pure[simp]: "rdom (pure R) = Range R"
  unfolding rdomp_def[abs_def] pure_def by auto

lemma Range_of_constraint_conv[simp]: "Range (A∩UNIV×C) = Range A ∩ C"
  by auto


subsection ‹Heap-Nres Refinement Calculus›

text ‹Predicate that expresses refinement. Given a heap
  ‹Γ›, program ‹c› produces a heap ‹Γ'› and
  a concrete result that is related with predicate ‹R› to some
  abstract result from ‹m››
definition "hn_refine Γ c Γ' R m ≡ nofail m ⟶
  <Γ> c <λr. Γ' * (∃⇩Ax. R x r * ↑(RETURN x ≤ m)) >⇩t"

(* TODO: Can we change the patterns of assn_simproc to add this pattern? *)
simproc_setup assn_simproc_hnr ("hn_refine Γ c Γ'")
  = ‹K Seplogic_Auto.assn_simproc_fun›

lemma hn_refineI[intro?]:
  assumes "nofail m 
    ⟹ <Γ> c <λr. Γ' * (∃⇩Ax. R x r * ↑(RETURN x ≤ m)) >⇩t"
  shows "hn_refine Γ c Γ' R m"
  using assms unfolding hn_refine_def by blast

lemma hn_refineD:
  assumes "hn_refine Γ c Γ' R m"
  assumes "nofail m"
  shows "<Γ> c <λr. Γ' * (∃⇩Ax. R x r * ↑(RETURN x ≤ m)) >⇩t"
  using assms unfolding hn_refine_def by blast

lemma hn_refine_preI: 
  assumes "⋀h. h⊨Γ ⟹ hn_refine Γ c Γ' R a"
  shows "hn_refine Γ c Γ' R a"
  using assms unfolding hn_refine_def
  by (auto intro: hoare_triple_preI)

lemma hn_refine_nofailI: 
  assumes "nofail a ⟹ hn_refine Γ c Γ' R a"  
  shows "hn_refine Γ c Γ' R a"
  using assms by (auto simp: hn_refine_def)

lemma hn_refine_false[simp]: "hn_refine false c Γ' R m"
  by rule auto

lemma hn_refine_fail[simp]: "hn_refine Γ c Γ' R FAIL"
  by rule auto

lemma hn_refine_frame:
  assumes "hn_refine P' c Q' R m"
  assumes "P ⟹⇩t F * P'"
  shows "hn_refine P c (F * Q') R m"
  using assms
  unfolding hn_refine_def entailst_def
  apply clarsimp
  apply (erule cons_pre_rule)
  apply (rule cons_post_rule)
  apply (erule fi_rule, frame_inference)
  apply (simp only: star_aci)
  apply simp
  done

lemma hn_refine_cons:
  assumes I: "P⟹⇩tP'"
  assumes R: "hn_refine P' c Q R m"
  assumes I': "Q⟹⇩tQ'"
  assumes R': "⋀x y. R x y ⟹⇩t R' x y"
  shows "hn_refine P c Q' R' m"
  using R unfolding hn_refine_def
  apply clarify
  apply (rule cons_pre_rulet[OF I])
  apply (rule cons_post_rulet)
  apply assumption
  apply (sep_auto simp: entailst_def)
  apply (rule enttD)
  apply (intro entt_star_mono I' R')
  done

(*lemma hn_refine_cons:
  assumes I: "P⟹⇩AP'"
  assumes R: "hn_refine P' c Q R m"
  assumes I': "Q⟹⇩AQ'"
  assumes R': "⋀x y. R x y ⟹⇩A R' x y"
  shows "hn_refine P c Q' R' m"
  using R unfolding hn_refine_def
  apply clarsimp
  apply (rule cons_pre_rule[OF I])
  apply (erule cons_post_rule)
  apply (rule ent_star_mono ent_refl I' R' ent_ex_preI ent_ex_postI)+
  done
*)
lemma hn_refine_cons_pre:
  assumes I: "P⟹⇩tP'"
  assumes R: "hn_refine P' c Q R m"
  shows "hn_refine P c Q R m"
  by (rule hn_refine_cons[OF I R]) sep_auto+

lemma hn_refine_cons_post:
  assumes R: "hn_refine P c Q R m"
  assumes I: "Q⟹⇩tQ'"
  shows "hn_refine P c Q' R m"
  using assms
  by (rule hn_refine_cons[OF entt_refl _ _ entt_refl])

lemma hn_refine_cons_res: 
  "⟦ hn_refine Γ f Γ' R g; ⋀a c. R a c ⟹⇩t R' a c ⟧ ⟹ hn_refine Γ f Γ' R' g"
  by (erule hn_refine_cons[OF entt_refl]) sep_auto+

lemma hn_refine_ref:
  assumes LE: "m≤m'"
  assumes R: "hn_refine P c Q R m"
  shows "hn_refine P c Q R m'"
  apply rule
  apply (rule cons_post_rule)
  apply (rule hn_refineD[OF R])
  using LE apply (simp add: pw_le_iff)
  apply (sep_auto intro: order_trans[OF _ LE])
  done

lemma hn_refine_cons_complete:
  assumes I: "P⟹⇩tP'"
  assumes R: "hn_refine P' c Q R m"
  assumes I': "Q⟹⇩tQ'"
  assumes R': "⋀x y. R x y ⟹⇩t R' x y"
  assumes LE: "m≤m'"
  shows "hn_refine P c Q' R' m'"
  apply (rule hn_refine_ref[OF LE])
  apply (rule hn_refine_cons[OF I R I' R'])
  done
 
lemma hn_refine_augment_res:
  assumes A: "hn_refine Γ f Γ' R g"
  assumes B: "g ≤⇩n SPEC Φ"
  shows "hn_refine Γ f Γ' (λa c. R a c * ↑(Φ a)) g"
  apply (rule hn_refineI)
  apply (rule cons_post_rule)
  apply (erule A[THEN hn_refineD])
  using B
  apply (sep_auto simp: pw_le_iff pw_leof_iff)
  done


subsection ‹Product Types›
text ‹Some notion for product types is already defined here, as it is used 
  for currying and uncurrying, which is fundamental for the sepref tool›
definition prod_assn :: "('a1⇒'c1⇒assn) ⇒ ('a2⇒'c2⇒assn) 
  ⇒ 'a1*'a2 ⇒ 'c1*'c2 ⇒ assn" where
  "prod_assn P1 P2 a c ≡ case (a,c) of ((a1,a2),(c1,c2)) ⇒
  P1 a1 c1 * P2 a2 c2"

notation prod_assn (infixr "×⇩a" 70)
  
lemma prod_assn_pure_conv[simp]: "prod_assn (pure R1) (pure R2) = pure (R1 ×⇩r R2)"
  by (auto simp: pure_def prod_assn_def intro!: ext)

lemma prod_assn_pair_conv[simp]: 
  "prod_assn A B (a1,b1) (a2,b2) = A a1 a2 * B b1 b2"
  unfolding prod_assn_def by auto

lemma prod_assn_true[simp]: "prod_assn (λ_ _. true) (λ_ _. true) = (λ_ _. true)"
  by (auto intro!: ext simp: hn_ctxt_def prod_assn_def)

subsection "Convenience Lemmas"

lemma hn_refine_guessI:
  assumes "hn_refine P f P' R f'"
  assumes "f=f_conc"
  shows "hn_refine P f_conc P' R f'"
  ― ‹To prove a refinement, first synthesize one, and then prove equality›
  using assms by simp


lemma imp_correctI:
  assumes R: "hn_refine Γ c Γ' R a"
  assumes C: "a ≤ SPEC Φ"
  shows "<Γ> c <λr'. ∃⇩Ar. Γ' * R r r' * ↑(Φ r)>⇩t"
  apply (rule cons_post_rule)
  apply (rule hn_refineD[OF R])
  apply (rule le_RES_nofailI[OF C])
  apply (sep_auto dest: order_trans[OF _ C])
  done

lemma hnr_pre_ex_conv: 
  shows "hn_refine (∃⇩Ax. Γ x) c Γ' R a ⟷ (∀x. hn_refine (Γ x) c Γ' R a)"
  unfolding hn_refine_def
  apply safe
  apply (erule cons_pre_rule[rotated])
  apply (rule ent_ex_postI)
  apply (rule ent_refl)
  apply sep_auto
  done

lemma hnr_pre_pure_conv:  
  shows "hn_refine (Γ * ↑P) c Γ' R a ⟷ (P ⟶ hn_refine Γ c Γ' R a)"
  unfolding hn_refine_def
  by auto

lemma hn_refine_split_post:
  assumes "hn_refine Γ c Γ' R a"
  shows "hn_refine Γ c (Γ' ∨⇩A Γ'') R a"
  apply (rule hn_refine_cons_post[OF assms])
  by (rule entt_disjI1_direct)

lemma hn_refine_post_other: 
  assumes "hn_refine Γ c Γ'' R a"
  shows "hn_refine Γ c (Γ' ∨⇩A Γ'') R a"
  apply (rule hn_refine_cons_post[OF assms])
  by (rule entt_disjI2_direct)


subsubsection ‹Return›

lemma hnr_RETURN_pass:
  "hn_refine (hn_ctxt R x p) (return p) (hn_invalid R x p) R (RETURN x)"
  ― ‹Pass on a value from the heap as return value›
  apply rule 
  apply (sep_auto simp: hn_ctxt_def eintros: invalidate_clone')
  done

lemma hnr_RETURN_pure:
  assumes "(c,a)∈R"
  shows "hn_refine emp (return c) emp (pure R) (RETURN a)"
  ― ‹Return pure value›
  unfolding hn_refine_def using assms
  by (sep_auto simp: pure_def)
  
subsubsection ‹Assertion›
lemma hnr_FAIL[simp, intro!]: "hn_refine Γ c Γ' R FAIL"
  unfolding hn_refine_def
  by simp

lemma hnr_ASSERT:
  assumes "Φ ⟹ hn_refine Γ c Γ' R c'"
  shows "hn_refine Γ c Γ' R (do { ASSERT Φ; c'})"
  using assms
  apply (cases Φ)
  by auto

subsubsection ‹Bind›
lemma bind_det_aux: "⟦ RETURN x ≤ m; RETURN y ≤ f x ⟧ ⟹ RETURN y ≤ m ⤜ f"
  apply (rule order_trans[rotated])
  apply (rule Refine_Basic.bind_mono)
  apply assumption
  apply (rule order_refl)
  apply simp
  done

lemma hnr_bind:
  assumes D1: "hn_refine Γ m' Γ1 Rh m"
  assumes D2: 
    "⋀x x'. RETURN x ≤ m ⟹ hn_refine (Γ1 * hn_ctxt Rh x x') (f' x') (Γ2 x x') R (f x)"
  assumes IMP: "⋀x x'. Γ2 x x' ⟹⇩t Γ' * hn_ctxt Rx x x'"
  shows "hn_refine Γ (m'⤜f') Γ' R (m⤜f)"
  using assms
  unfolding hn_refine_def
  apply (clarsimp simp add: pw_bind_nofail)
  apply (rule Hoare_Triple.bind_rule)
  apply assumption
  apply (clarsimp intro!: normalize_rules simp: hn_ctxt_def)
proof -
  fix x' x
  assume 1: "RETURN x ≤ m" 
    and "nofail m" "∀x. inres m x ⟶ nofail (f x)"
  hence "nofail (f x)" by (auto simp: pw_le_iff)
  moreover assume "⋀x x'. RETURN x ≤ m ⟹
           nofail (f x) ⟶ <Γ1 * Rh x x'> f' x'
           <λr'. ∃⇩Ar. Γ2 x x' * R r r' * true * ↑ (RETURN r ≤ f x)>"
  ultimately have "⋀x'. <Γ1 * Rh x x'> f' x'
           <λr'. ∃⇩Ar. Γ2 x x' * R r r' * true * ↑ (RETURN r ≤ f x)>"
    using 1 by simp
  also have "⋀r'. ∃⇩Ar. Γ2 x x' * R r r' * true * ↑ (RETURN r ≤ f x) ⟹⇩A
    ∃⇩Ar. Γ' * R r r' * true * ↑ (RETURN r ≤ f x)"
    apply (sep_auto)
    apply (rule ent_frame_fwd[OF IMP[THEN enttD]])
    apply frame_inference
    apply (solve_entails)
    done
  finally (cons_post_rule) have 
    R: "<Γ1 * Rh x x'> f' x' 
        <λr'. ∃⇩Ar. Γ' * R r r' * true * ↑(RETURN r ≤ f x)>"
    .
  show "<Γ1 * Rh x x' * true> f' x'
          <λr'. ∃⇩Ar. Γ' * R r r' * true * ↑ (RETURN r ≤ m ⤜ f)>"
    by (sep_auto heap: R intro: bind_det_aux[OF 1])
qed

subsubsection ‹Recursion›

definition "hn_rel P m ≡ λr. ∃⇩Ax. P x r * ↑(RETURN x ≤ m)"

lemma hn_refine_alt: "hn_refine Fpre c Fpost P m ≡ nofail m ⟶
  <Fpre> c <λr. hn_rel P m r * Fpost>⇩t"
  apply (rule eq_reflection)
  unfolding hn_refine_def hn_rel_def
  apply (simp add: hn_ctxt_def)
  apply (simp only: star_aci)
  done

lemma wit_swap_forall:
  assumes W: "<P> c <λ_. true>"
  assumes T: "(∀x. A x ⟶ <P> c <Q x>)"
  shows "<P> c <λr. ¬⇩A (∃⇩Ax. ↑(A x) * ¬⇩A Q x r)>"
  unfolding hoare_triple_def Let_def
  apply (intro conjI impI allI)
  subgoal by (elim conjE) (rule hoare_tripleD[OF W], assumption+) []

  subgoal
    apply (clarsimp, intro conjI allI)
    apply1 (rule models_in_range)
    applyS (rule hoare_tripleD[OF W]; assumption; fail)
    apply1 (simp only: disj_not2, intro impI)
    apply1 (drule spec[OF T, THEN mp])
    apply1 (drule (2) hoare_tripleD(2))
    by assumption

  subgoal by (elim conjE) (rule hoare_tripleD[OF W], assumption+)

  subgoal by (elim conjE) (rule hoare_tripleD[OF W], assumption+) 
  done

lemma hn_admissible:
  assumes PREC: "precise Ry"
  assumes E: "∀f∈A. nofail (f x) ⟶ <P> c <λr. hn_rel Ry (f x) r * F>"
  assumes NF: "nofail (INF f∈A. f x)"
  shows "<P> c <λr. hn_rel Ry (INF f∈A. f x) r * F>"
proof -
  from NF obtain f where "f∈A" and "nofail (f x)"
    by (simp only: refine_pw_simps) blast

  with E have "<P> c <λr. hn_rel Ry (f x) r * F>" by blast
  hence W: "<P> c <λ_. true>" by (rule cons_post_rule, simp)

  from E have 
    E': "∀f. f∈A ∧ nofail (f x) ⟶ <P> c <λr. hn_rel Ry (f x) r * F>"
    by blast
  from wit_swap_forall[OF W E'] have 
    E'': "<P> c
     <λr. ¬⇩A (∃⇩Axa. ↑ (xa ∈ A ∧ nofail (xa x)) *
                ¬⇩A (hn_rel Ry (xa x) r * F))>" .
  
  thus ?thesis
    apply (rule cons_post_rule)
    unfolding entails_def hn_rel_def
    apply clarsimp
  proof -
    fix h as p
    assume A: "∀f. f∈A ⟶ (∃a.
      ((h, as) ⊨ Ry a p * F ∧ RETURN a ≤ f x)) ∨ ¬ nofail (f x)"
    with ‹f∈A› and ‹nofail (f x)› obtain a where 
      1: "(h, as) ⊨ Ry a p * F" and "RETURN a ≤ f x"
      by blast
    have
      "∀f∈A. nofail (f x) ⟶ (h, as) ⊨ Ry a p * F ∧ RETURN a ≤ f x"
    proof clarsimp
      fix f'
      assume "f'∈A" and "nofail (f' x)"
      with A obtain a' where 
        2: "(h, as) ⊨ Ry a' p * F" and "RETURN a' ≤ f' x"
        by blast

      moreover note preciseD'[OF PREC 1 2] 
      ultimately show "(h, as) ⊨ Ry a p * F ∧ RETURN a ≤ f' x" by simp
    qed
    hence "RETURN a ≤ (INF f∈A. f x)"
      by (metis (mono_tags) le_INF_iff le_nofailI)
    with 1 show "∃a. (h, as) ⊨ Ry a p * F ∧ RETURN a ≤ (INF f∈A. f x)"
      by blast
  qed
qed

lemma hn_admissible':
  assumes PREC: "precise Ry"
  assumes E: "∀f∈A. nofail (f x) ⟶ <P> c <λr. hn_rel Ry (f x) r * F>⇩t"
  assumes NF: "nofail (INF f∈A. f x)"
  shows "<P> c <λr. hn_rel Ry (INF f∈A. f x) r * F>⇩t"
  apply (rule hn_admissible[OF PREC, where F="F*true", simplified])
  apply simp
  by fact+

lemma hnr_RECT_old:
  assumes S: "⋀cf af ax px. ⟦
    ⋀ax px. hn_refine (hn_ctxt Rx ax px * F) (cf px) (F' ax px) Ry (af ax)⟧ 
    ⟹ hn_refine (hn_ctxt Rx ax px * F) (cB cf px) (F' ax px) Ry (aB af ax)"
  assumes M: "(⋀x. mono_Heap (λf. cB f x))"
  assumes PREC: "precise Ry"
  shows "hn_refine 
    (hn_ctxt Rx ax px * F) (heap.fixp_fun cB px) (F' ax px) Ry (RECT aB ax)"
  unfolding RECT_gfp_def
proof (simp, intro conjI impI)
  assume "trimono aB"
  hence "mono aB" by (simp add: trimonoD)
  have "∀ax px. 
    hn_refine (hn_ctxt Rx ax px * F) (heap.fixp_fun cB px) (F' ax px) Ry 
      (gfp aB ax)"
    apply (rule gfp_cadm_induct[OF _ _ ‹mono aB›])

    apply rule
    apply (auto simp: hn_refine_alt intro: hn_admissible'[OF PREC]) []

    apply (auto simp: hn_refine_alt) []

    apply clarsimp
    apply (subst heap.mono_body_fixp[of cB, OF M])
    apply (rule S)
    apply blast
    done
  thus "hn_refine (hn_ctxt Rx ax px * F)
     (ccpo.fixp (fun_lub Heap_lub) (fun_ord Heap_ord) cB px) (F' ax px) Ry
     (gfp aB ax)" by simp
qed

lemma hnr_RECT:
  assumes S: "⋀cf af ax px. ⟦
    ⋀ax px. hn_refine (hn_ctxt Rx ax px * F) (cf px) (F' ax px) Ry (af ax)⟧ 
    ⟹ hn_refine (hn_ctxt Rx ax px * F) (cB cf px) (F' ax px) Ry (aB af ax)"
  assumes M: "(⋀x. mono_Heap (λf. cB f x))"
  shows "hn_refine 
    (hn_ctxt Rx ax px * F) (heap.fixp_fun cB px) (F' ax px) Ry (RECT aB ax)"
  unfolding RECT_def
proof (simp, intro conjI impI)
  assume "trimono aB"
  hence "flatf_mono_ge aB" by (simp add: trimonoD)
  have "∀ax px. 
    hn_refine (hn_ctxt Rx ax px * F) (heap.fixp_fun cB px) (F' ax px) Ry 
      (flatf_gfp aB ax)"
      
    apply (rule flatf_ord.fixp_induct[OF _ ‹flatf_mono_ge aB›])  

    apply (rule flatf_admissible_pointwise)
    apply simp

    apply (auto simp: hn_refine_alt) []

    apply clarsimp
    apply (subst heap.mono_body_fixp[of cB, OF M])
    apply (rule S)
    apply blast
    done
  thus "hn_refine (hn_ctxt Rx ax px * F)
     (ccpo.fixp (fun_lub Heap_lub) (fun_ord Heap_ord) cB px) (F' ax px) Ry
     (flatf_gfp aB ax)" by simp
qed

lemma hnr_If:
  assumes P: "Γ ⟹⇩t Γ1 * hn_val bool_rel a a'"
  assumes RT: "a ⟹ hn_refine (Γ1 * hn_val bool_rel a a') b' Γ2b R b"
  assumes RE: "¬a ⟹ hn_refine (Γ1 * hn_val bool_rel a a') c' Γ2c R c"
  assumes IMP: "Γ2b ∨⇩A Γ2c ⟹⇩t Γ'"
  shows "hn_refine Γ (if a' then b' else c') Γ' R (if a then b else c)"
  apply (rule hn_refine_cons[OF P])
  apply1 (rule hn_refine_preI)
  applyF (cases a; simp add: hn_ctxt_def pure_def)
    focus
      apply1 (rule hn_refine_split_post)
      applyF (rule hn_refine_cons_pre[OF _ RT])
        applyS (simp add: hn_ctxt_def pure_def)
        applyS simp
      solved
    solved
    apply1 (rule hn_refine_post_other)
    applyF (rule hn_refine_cons_pre[OF _ RE])
      applyS (simp add: hn_ctxt_def pure_def)
      applyS simp
    solved
  solved
  applyS (rule IMP)
  applyS (rule entt_refl)
  done


subsection ‹ML-Level Utilities›
ML ‹
  signature SEPREF_BASIC = sig
    (* Destroy lambda term, return function to reconstruct. Bound var is replaced by free. *)
    val dest_lambda_rc: Proof.context -> term -> ((term * (term -> term)) * Proof.context)
    (* Apply function under lambda. Bound var is replaced by free. *)
    val apply_under_lambda: (Proof.context -> term -> term) -> Proof.context -> term -> term

    (* 'a nres type *)
    val is_nresT: typ -> bool
    val mk_nresT: typ -> typ
    val dest_nresT: typ -> typ

    (* Make certified == *)
    val mk_cequals: cterm * cterm -> cterm
    (* Make ⟹⇩A *)
    val mk_entails: term * term -> term


    (* Operations on pre-terms *)
    val constrain_type_pre: typ -> term -> term (* t::T *)

    val mk_pair_in_pre: term -> term -> term -> term (* (c,a) ∈ R *)

    val mk_compN_pre: int -> term -> term -> term  (* f o...o g*)

    val mk_curry0_pre: term -> term                (* curry0 f *) 
    val mk_curry_pre: term -> term                 (* curry f *) 
    val mk_curryN_pre: int -> term -> term         (* curry (...(curry f)...) *) 

    val mk_uncurry0_pre: term -> term              (* uncurry0 f *)       
    val mk_uncurry_pre: term -> term               (* uncurry f *)
    val mk_uncurryN_pre: int -> term -> term       (* uncurry (...(uncurry f)...) *)



    (* Conversion for hn_refine - term*)
    val hn_refine_conv: conv -> conv -> conv -> conv -> conv -> conv

    (* Conversion on abstract value (last argument) of hn_refine - term *)
    val hn_refine_conv_a: conv -> conv

    (* Conversion on abstract value of hn_refine term in conclusion of theorem *)
    val hn_refine_concl_conv_a: (Proof.context -> conv) -> Proof.context -> conv

    (* Destruct hn_refine term *)
    val dest_hn_refine: term -> term * term * term * term * term 
    (* Make hn_refine term *)
    val mk_hn_refine: term * term * term * term * term -> term
    (* Check if given term is Trueprop (hn_refine ...). Use with CONCL_COND'. *)
    val is_hn_refine_concl: term -> bool

    (* Destruct abs-fun, returns RETURN-flag, (f, args) *)
    val dest_hnr_absfun: term -> bool * (term * term list)
    (* Make abs-fun. *)
    val mk_hnr_absfun: bool * (term * term list) -> term
    (* Make abs-fun. Guess RETURN-flag from type. *)
    val mk_hnr_absfun': (term * term list) -> term
    
    (* Prove permutation of *. To be used with f_tac_conv. *)
    val star_permute_tac: Proof.context -> tactic

    (* Make separation conjunction *)
    val mk_star: term * term -> term
    (* Make separation conjunction from list. "[]" yields "emp". *)
    val list_star: term list -> term
    (* Decompose separation conjunction. "emp" yields "[]". *)
    val strip_star: term -> term list

    (* Check if true-assertion *)
    val is_true: term -> bool

    (* Check if term is hn_ctxt-assertion *)
    val is_hn_ctxt: term -> bool 
    (* Decompose hn_ctxt-assertion *)
    val dest_hn_ctxt: term -> term * term * term
    (* Decompose hn_ctxt-assertion, NONE if term has wrong format *)
    val dest_hn_ctxt_opt: term -> (term * term * term) option
      

    type phases_ctrl = {
      trace: bool,            (* Trace phases *)
      int_res: bool,          (* Stop with intermediate result *)
      start: string option,   (* Start with this phase. NONE: First phase *)
      stop: string option     (* Stop after this phase. NONE: Last phase *)
    }

    (* No tracing or intermediate result, all phases *)
    val dflt_phases_ctrl: phases_ctrl 
    (* Tracing, intermediate result, all phases *)
    val dbg_phases_ctrl: phases_ctrl
    val flag_phases_ctrl: bool -> phases_ctrl

    (* Name, tactic, expected number of created goals (may be negative for solved goals) *)
    type phase = string * (Proof.context -> tactic') * int

    (* Perform sequence of tactics (tac,n), each expected to create n new goals, 
       or solve goals if n is negative. 
       Debug-flag: Stop with intermediate state after tactic 
       fails or produces less/more goals as expected. *)   
    val PHASES': phase list -> phases_ctrl -> Proof.context -> tactic'

  end

  structure Sepref_Basic: SEPREF_BASIC = struct

    fun is_nresT (Type (@{type_name nres},[_])) = true | is_nresT _ = false
    fun mk_nresT T = Type(@{type_name nres},[T])
    fun dest_nresT (Type (@{type_name nres},[T])) = T | dest_nresT T = raise TYPE("dest_nresT",[T],[])


    fun dest_lambda_rc ctxt (Abs (x,T,t)) = let
        val (u,ctxt) = yield_singleton Variable.variant_fixes x ctxt
        val u = Free (u,T)
        val t = subst_bound (u,t)
        val reconstruct = Term.lambda_name (x,u)
      in
        ((t,reconstruct),ctxt)
      end
    | dest_lambda_rc _ t = raise TERM("dest_lambda_rc",[t])

    fun apply_under_lambda f ctxt t = let
      val ((t,rc),ctxt) = dest_lambda_rc ctxt t
      val t = f ctxt t
    in
      rc t
    end


    (* Functions on pre-terms *)
    fun mk_pair_in_pre x y r = Const (@{const_name Set.member}, dummyT) $
      (Const (@{const_name Product_Type.Pair}, dummyT) $ x $ y) $ r


    fun mk_uncurry_pre t = Const(@{const_name uncurry}, dummyT)$t
    fun mk_uncurry0_pre t = Const(@{const_name uncurry0}, dummyT)$t
    fun mk_uncurryN_pre 0 = mk_uncurry0_pre
      | mk_uncurryN_pre 1 = I
      | mk_uncurryN_pre n = mk_uncurry_pre o mk_uncurryN_pre (n-1)

    fun mk_curry_pre t = Const(@{const_name curry}, dummyT)$t
    fun mk_curry0_pre t = Const(@{const_name curry0}, dummyT)$t
    fun mk_curryN_pre 0 = mk_curry0_pre
      | mk_curryN_pre 1 = I
      | mk_curryN_pre n = mk_curry_pre o mk_curryN_pre (n-1)


    fun mk_compN_pre 0 f g = f $ g
      | mk_compN_pre n f g = let
          val g = fold (fn i => fn t => t$Bound i) (n-2 downto 0) g
          val t = Const(@{const_name "Fun.comp"},dummyT) $ f $ g

          val t = fold (fn i => fn t => Abs ("x"^string_of_int i,dummyT,t)) (n-1 downto 1) t
        in
          t
        end

    fun constrain_type_pre T t = Const(@{syntax_const "_type_constraint_"},T-->T) $ t




    local open Conv in
      fun hn_refine_conv c1 c2 c3 c4 c5 ct = case Thm.term_of ct of
        @{mpat "hn_refine _ _ _ _ _"} => let
          val cc = combination_conv
        in
          cc (cc (cc (cc (cc all_conv c1) c2) c3) c4) c5 ct
        end
      | _ => raise CTERM ("hn_refine_conv",[ct])
  
      val hn_refine_conv_a = hn_refine_conv all_conv all_conv all_conv all_conv
  
      fun hn_refine_concl_conv_a conv ctxt = Refine_Util.HOL_concl_conv 
        (fn ctxt => hn_refine_conv_a (conv ctxt)) ctxt
  
    end

    (* FIXME: Strange dependency! *)
    val mk_cequals = uncurry SMT_Util.mk_cequals
  
    val mk_entails = HOLogic.mk_binrel @{const_name "entails"}
  
    val mk_star = HOLogic.mk_binop @{const_name "Groups.times_class.times"}

    fun list_star [] = @{term "emp::assn"}
      | list_star [a] = a
      | list_star (a::l) = mk_star (list_star l,a)

    fun strip_star @{mpat "?a*?b"} = strip_star a @ strip_star b
      | strip_star @{mpat "emp"} = []
      | strip_star t = [t]

    fun is_true @{mpat "true"} = true | is_true _ = false
  
    fun is_hn_ctxt @{mpat "hn_ctxt _ _ _"} = true | is_hn_ctxt _ = false
    fun dest_hn_ctxt @{mpat "hn_ctxt ?R ?a ?p"} = (R,a,p) 
      | dest_hn_ctxt t = raise TERM("dest_hn_ctxt",[t])
  
    fun dest_hn_ctxt_opt @{mpat "hn_ctxt ?R ?a ?p"} = SOME (R,a,p) 
      | dest_hn_ctxt_opt _ = NONE
  
    fun strip_abs_args (t as @{mpat "PR_CONST _"}) = (t,[])
      | strip_abs_args @{mpat "?f$?a"} = (case strip_abs_args f of (f,args) => (f,args@[a]))
      | strip_abs_args t = (t,[])
  
    fun dest_hnr_absfun @{mpat "RETURN$?a"} = (true, strip_abs_args a)
      | dest_hnr_absfun f = (false, strip_abs_args f)
  
    fun mk_hnr_absfun (true,fa) = Autoref_Tagging.list_APP fa |> (fn a => @{mk_term "RETURN$?a"})
      | mk_hnr_absfun (false,fa) = Autoref_Tagging.list_APP fa
  
    fun mk_hnr_absfun' fa = let
      val t = Autoref_Tagging.list_APP fa
      val T = fastype_of t
    in
      case T of
        Type (@{type_name nres},_) => t
      | _ => @{mk_term "RETURN$?t"}
  
    end  
  
    fun dest_hn_refine @{mpat "hn_refine ?P ?c ?Q ?R ?a"} = (P,c,Q,R,a)
      | dest_hn_refine t = raise TERM("dest_hn_refine",[t])
  
    fun mk_hn_refine (P,c,Q,R,a) = @{mk_term "hn_refine ?P ?c ?Q ?R ?a"}
  
    val is_hn_refine_concl = can (HOLogic.dest_Trueprop #> dest_hn_refine)
  
    fun star_permute_tac ctxt = ALLGOALS (simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms star_aci}))
      

    type phases_ctrl = {
      trace: bool,            
      int_res: bool,          
      start: string option,   
      stop: string option     
    }

    val dflt_phases_ctrl = {trace=false,int_res=false,start=NONE,stop=NONE} 
    val dbg_phases_ctrl = {trace=true,int_res=true,start=NONE,stop=NONE}
    fun flag_phases_ctrl dbg = if dbg then dbg_phases_ctrl else dflt_phases_ctrl

    type phase = string * (Proof.context -> tactic') * int

    local
      fun ph_range phases start stop = let
        fun find_phase name = let
          val i = find_index (fn (n,_,_) => n=name) phases
          val _ = if i<0 then error ("No such phase: " ^ name) else ()
        in
          i
        end

        val i = case start of NONE => 0 | SOME n => find_phase n
        val j = case stop of NONE => length phases - 1 | SOME n => find_phase n

        val phases = take (j+1) phases |> drop i

        val _ = case phases of [] => error "No phases selected, range is empty" | _ => ()
      in
        phases
      end
    in  
  
      fun PHASES' phases ctrl ctxt = let
        val phases = ph_range phases (#start ctrl) (#stop ctrl)
        val phases = map (fn (n,tac,d) => (n,tac ctxt,d)) phases
  
        fun r [] _ st = Seq.single st
          | r ((name,tac,d)::tacs) i st = let
              val n = Thm.nprems_of st
              val bailout_tac = if #int_res ctrl then all_tac else no_tac
              fun trace_tac msg st = (if #trace ctrl then tracing msg else (); Seq.single st)
              val trace_start_tac = trace_tac ("Phase " ^ name)
            in
              K trace_start_tac THEN' IF_EXGOAL (tac)
              THEN_ELSE' (
                fn i => fn st => 
                  (* Bail out if a phase does not solve/create exactly the expected subgoals *)
                  if Thm.nprems_of st = n+d then
                    ((trace_tac "  Done" THEN r tacs i) st)
                  else
                    (trace_tac "*** Wrong number of produced goals" THEN bailout_tac) st
              , 
                K (trace_tac "*** Phase tactic failed" THEN bailout_tac))
            end i st
  
      in
        r phases
      end


    end

(*    (* Perform sequence of tactics (tac,n), each expected to create n new goals, 
       or solve goals if n is negative. 
       Debug-flag: Stop with intermediate state after tactic 
       fails or produces less/more goals as expected. *)   
    val PHASES': phase list -> phases_ctrl -> Proof.context -> tactic'
*)



(*

    fun xPHASES' dbg tacs ctxt = let
      val tacs = map (fn (tac,d) => (tac ctxt,d)) tacs

      fun r [] _ st = Seq.single st
        | r ((tac,d)::tacs) i st = let
            val n = Thm.nprems_of st
            val bailout_tac = if dbg then all_tac else no_tac
          in
            IF_EXGOAL (tac)
            THEN_ELSE' (
              fn i => fn st => 
                (* Bail out if a phase does not solve/create exactly the expected subgoals *)
                if Thm.nprems_of st = n+d then
                  (r tacs i st)
                else
                  bailout_tac st
            , 
              K bailout_tac)
          end i st

    in
      r tacs
    end
*)
  end


  signature SEPREF_DEBUGGING = sig
    (*************************)
    (* Debugging *)
    (* Centralized debugging mode flag *)
    val cfg_debug_all: bool Config.T

    val is_debug: bool Config.T -> Proof.context -> bool
    val is_debug': Proof.context -> bool

    (* Conversion, trace errors if custom or central debugging flag is activated *)
    val DBG_CONVERSION: bool Config.T -> Proof.context -> conv -> tactic'

    (* Conversion, trace errors if central debugging flag is activated *)
    val DBG_CONVERSION': Proof.context -> conv -> tactic'

    (* Tracing message and current subgoal *)
    val tracing_tac': string -> Proof.context -> tactic'
    (* Warning message and current subgoal *)
    val warning_tac': string -> Proof.context -> tactic'
    (* Error message and current subgoal *)
    val error_tac': string -> Proof.context -> tactic'

    (* Trace debug message *)
    val dbg_trace_msg: bool Config.T -> Proof.context -> string -> unit
    val dbg_trace_msg': Proof.context -> string -> unit

    val dbg_msg_tac: bool Config.T -> (Proof.context -> int -> thm -> string) -> Proof.context -> tactic'
    val dbg_msg_tac': (Proof.context -> int -> thm -> string) -> Proof.context -> tactic'

    val msg_text: string -> Proof.context -> int -> thm -> string
    val msg_subgoal: string -> Proof.context -> int -> thm -> string
    val msg_from_subgoal: string -> (term -> Proof.context -> string) -> Proof.context -> int -> thm -> string
    val msg_allgoals: string -> Proof.context -> int -> thm -> string

  end

  structure Sepref_Debugging: SEPREF_DEBUGGING = struct

    val cfg_debug_all = 
      Attrib.setup_config_bool @{binding sepref_debug_all} (K false)

    fun is_debug cfg ctxt = Config.get ctxt cfg orelse Config.get ctxt cfg_debug_all
    fun is_debug' ctxt = Config.get ctxt cfg_debug_all

    fun dbg_trace cfg ctxt obj = 
      if is_debug cfg ctxt then  
        tracing (@{make_string} obj)
      else ()

    fun dbg_trace' ctxt obj = 
      if is_debug' ctxt then  
        tracing (@{make_string} obj)
      else ()

    fun dbg_trace_msg cfg ctxt msg =   
      if is_debug cfg ctxt then  
        tracing msg
      else ()
    fun dbg_trace_msg' ctxt msg = 
      if is_debug' ctxt then  
        tracing msg
      else ()

    fun DBG_CONVERSION cfg ctxt cv i st = 
      Seq.single (Conv.gconv_rule cv i st)
      handle e as THM _ => (dbg_trace cfg ctxt e; Seq.empty)
           | e as CTERM _ => (dbg_trace cfg ctxt e; Seq.empty)
           | e as TERM _ => (dbg_trace cfg ctxt e; Seq.empty)
           | e as TYPE _ => (dbg_trace cfg ctxt e; Seq.empty);

    fun DBG_CONVERSION' ctxt cv i st = 
      Seq.single (Conv.gconv_rule cv i st)
      handle e as THM _ => (dbg_trace' ctxt e; Seq.empty)
           | e as CTERM _ => (dbg_trace' ctxt e; Seq.empty)
           | e as TERM _ => (dbg_trace' ctxt e; Seq.empty)
           | e as TYPE _ => (dbg_trace' ctxt e; Seq.empty);


    local 
      fun gen_subgoal_msg_tac do_msg msg ctxt = IF_EXGOAL (fn i => fn st => let
        val t = nth (Thm.prems_of st) (i-1)
        val _ = Pretty.block [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt t]
          |> Pretty.string_of |> do_msg

      in
        Seq.single st
      end)
    in       
      val tracing_tac' = gen_subgoal_msg_tac tracing
      val warning_tac' = gen_subgoal_msg_tac warning
      val error_tac' = gen_subgoal_msg_tac error
    end


    fun dbg_msg_tac cfg msg ctxt =
      if is_debug cfg ctxt then (fn i => fn st => (tracing (msg ctxt i st); Seq.single st))
      else K all_tac
    fun dbg_msg_tac' msg ctxt =
      if is_debug' ctxt then (fn i => fn st => (tracing (msg ctxt i st); Seq.single st))
      else K all_tac

    fun msg_text msg _ _ _ = msg

    fun msg_from_subgoal msg sgmsg ctxt i st = 
      case try (nth (Thm.prems_of st)) (i-1) of
        NONE => msg ^ "\n" ^ "Subgoal out of range"
      | SOME t => msg ^ "\n" ^ sgmsg t ctxt

    fun msg_subgoal msg = msg_from_subgoal msg (fn t => fn ctxt =>
      Syntax.pretty_term ctxt t |> Pretty.string_of
    )

    fun msg_allgoals msg ctxt _ st = 
      msg ^ "\n" ^ Pretty.string_of (Pretty.chunks (Goal_Display.pretty_goals ctxt st))

  end
›


ML ‹
  (* Tactics for produced subgoals *)
  infix 1 THEN_NEXT THEN_ALL_NEW_LIST THEN_ALL_NEW_LIST'
  signature STACTICAL = sig
    (* Apply first tactic on this subgoal, and then second tactic on next subgoal *)
    val THEN_NEXT: tactic' * tactic' -> tactic'
    (* Apply tactics to the current and following subgoals *)
    val APPLY_LIST: tactic' list -> tactic'
    (* Apply list of tactics on subgoals emerging from tactic. 
      Requires exactly one tactic per emerging subgoal.*)
    val THEN_ALL_NEW_LIST: tactic' * tactic' list -> tactic'
    (* Apply list of tactics to subgoals emerging from tactic, use fallback for additional subgoals. *)
    val THEN_ALL_NEW_LIST': tactic' * (tactic' list * tactic') -> tactic'

  end

  structure STactical : STACTICAL = struct
    infix 1 THEN_WITH_GOALDIFF
    fun (tac1 THEN_WITH_GOALDIFF tac2) st = let
      val n1 = Thm.nprems_of st
    in
      st |> (tac1 THEN (fn st => tac2 (Thm.nprems_of st - n1) st ))
    end

    fun (tac1 THEN_NEXT tac2) i = 
      tac1 i THEN_WITH_GOALDIFF (fn d => (
        if d < ~1 then 
          (error "THEN_NEXT: Tactic solved more than one goal"; no_tac) 
        else 
          tac2 (i+1+d)
      ))

    fun APPLY_LIST [] = K all_tac
      | APPLY_LIST (tac::tacs) = tac THEN_NEXT APPLY_LIST tacs
            
    fun (tac1 THEN_ALL_NEW_LIST tacs) i = 
      tac1 i 
      THEN_WITH_GOALDIFF (fn d =>
        if d+1 <> length tacs then (
          error "THEN_ALL_NEW_LIST: Tactic produced wrong number of goals"; no_tac
        ) else APPLY_LIST tacs i
      )

    fun (tac1 THEN_ALL_NEW_LIST' (tacs,rtac)) i =  
      tac1 i 
      THEN_WITH_GOALDIFF (fn d => let
        val _ = if d+1 < length tacs then error "THEN_ALL_NEW_LIST': Tactic produced too few goals" else ();
        val tacs' = tacs @ replicate (d + 1 - length tacs) rtac
      in    
        APPLY_LIST tacs' i
      end)


  end


  open STactical
›

end